Re: "How can I make iPhone 14 Plus ring through a connected sleep headband instead of the handset?"
You may want to browse through the methods below, to see if one or more of the suggestions there may work for you, with your particular brand of sleep headband.
Wondering if Apple Support's method {link below} will be suitable for you:
See how to: Route and automatically answer calls on iPhone - Apple Support
Eg: says:
You can automatically direct the audio of phone or FaceTime calls to the iPhone speaker, a Bluetooth® headset, or your hearing device. iPhone can also automatically answer calls after a length of time you set.
- Go to Settings
> Accessibility > Touch > Call Audio Routing, then choose an audio destination. - Tap Auto-Answer Calls, turn on Auto-Answer Calls, then set the duration of time before the call is answered by tapping
or 
.
During a call, you can switch the audio routing from your hearing aid to the iPhone speaker by removing the hearing aid from your ear.
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Seems Siri can announce calls and message notifications on supported headphones:
Change Siri accessibility settings on iPhone - Apple Support
Eg: says: "Announce calls and notifications for specific apps
You can have Siri announce notifications and calls from apps like Messages on your iPhone speaker, supported headphones, when using CarPlay, and on MFi hearing devices.
- Go to Settings > Accessibility > Siri, then turn on Announce Notifications on Speaker.
- Tap Announce Notifications, then for each app, choose whether to announce all notifications or only time-sensitive notifications.
If you wish to use the above method: See the above article for the settings / how to / end calls with Siri.
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More info:
Announce calls, messages, and more with Siri on iPhone - Apple Support
